\(\int x^{100} \Gamma (-2,a x) \, dx\) [42]

Optimal result

Integrand size = 9, antiderivative size = 25 \[ \int x^{100} \Gamma (-2,a x) \, dx=\frac {1}{101} x^{101} \Gamma (-2,a x)-\frac {\Gamma (99,a x)}{101 a^{101}} \] Output:

1/101*x^99/a^2*Ei(3,a*x)-1/101*GAMMA(99,a*x)/a^101
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (-2,a x) \, dx=\frac {1}{101} x^{101} \Gamma (-2,a x)-\frac {\Gamma (99,a x)}{101 a^{101}} \] Input:

Integrate[x^100*Gamma[-2, a*x],x]
 

Output:

(x^101*Gamma[-2, a*x])/101 - Gamma[99, a*x]/(101*a^101)
 

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7116}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^{100} \Gamma (-2,a x) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma (-2,a x)-\frac {\Gamma (99,a x)}{101 a^{101}}\)

Input:

Int[x^100*Gamma[-2, a*x],x]
 

Output:

(x^101*Gamma[-2, a*x])/101 - Gamma[99, a*x]/(101*a^101)
 

Defintions of rubi rules used

rule 7116

Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 
1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b 
*(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(865\) vs. \(2(24)=48\).

Time = 0.06 (sec) , antiderivative size = 866, normalized size of antiderivative = 34.64

\[\text {Expression too large to display}\]

Input:

int(x^98/a^2*Ei(3,x*a),x)
 

Output:

1/a^101*(-1/202*(Psi(101)+gamma-3/2-Psi(102)+ln(x)+ln(a))*x^101*a^101-305/ 
40804*x^101*a^101+94268904488832477456261857430572424738096937640789516634 
94238777294707070023223798882976159207729119823605850588608460429412647567 
360000000000000000000000/101-1/61812*(306*a^100*x^100-306*a^99*x^99+612*a^ 
98*x^98+59976*a^97*x^97+5817672*a^96*x^96+558496512*a^95*x^95+53057168640* 
a^94*x^94+4987373852160*a^93*x^93+463825768250880*a^92*x^92+42671970679080 
960*a^91*x^91+3883149331796367360*a^90*x^90+349483439861673062400*a^89*x^8 
9+31104026147688902553600*a^88*x^88+2737154300996623424716800*a^87*x^87+23 
8132424186706237950361600*a^86*x^86+20479388480056736463731097600*a^85*x^8 
5+1740748020804822599417143296000*a^84*x^84+146222833747605098351040036864 
000*a^83*x^83+12136495201051223163136323059712000*a^82*x^82+99519260648620 
0299377178490896384000*a^81*x^81+80610601125382224249551457762607104000*a^ 
80*x^80+6448848090030577939964116621008568320000*a^79*x^79+509458999112415 
657257165213059676897280000*a^78*x^78+397378019307684212660588866186547979 
87840000*a^77*x^77+3059810748669168437486534269636419445063680000*a^76*x^7 
6+232545616898856801248976604492367877824839680000*a^75*x^75+1744092126741 
4260093673245336927590836862976000000*a^74*x^74+12906281737886552469318201 
54932641721927860224000000*a^73*x^73+9421585668657183302602287131008284570 
0733796352000000*a^72*x^72+67835416814331719778736467343259648904528333373 
44000000*a^71*x^71+4816314593817552104290289181371435072221511669514240...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (21) = 42\).

Time = 0.12 (sec) , antiderivative size = 806, normalized size of antiderivative = 32.24 \[ \int x^{100} \Gamma (-2,a x) \, dx=\text {Too large to display} \] Input:

integrate(x^100*gamma(-2,a*x),x, algorithm="fricas")
 

Output:

1/101*(a^101*x^101*gamma(-2, a*x) - (a^98*x^98 + 98*a^97*x^97 + 9506*a^96* 
x^96 + 912576*a^95*x^95 + 86694720*a^94*x^94 + 8149303680*a^93*x^93 + 7578 
85242240*a^92*x^92 + 69725442286080*a^91*x^91 + 6345015248033280*a^90*x^90 
 + 571051372322995200*a^89*x^89 + 50823572136746572800*a^88*x^88 + 4472474 
348033698406400*a^87*x^87 + 389105268278931761356800*a^86*x^86 + 334630530 
71988131476684800*a^85*x^85 + 2844359511118991175518208000*a^84*x^84 + 238 
926198933995258743529472000*a^83*x^83 + 19830874511521606475712946176000*a 
^82*x^82 + 1626131709944771731008461586432000*a^81*x^81 + 1317166685055265 
10211685388500992000*a^80*x^80 + 10537333480442120816934831080079360000*a^ 
79*x^79 + 832449344954927544537851655326269440000*a^78*x^78 + 649310489064 
84348473952429115449016320000*a^77*x^77 + 49996907657992948324943370418895 
74256640000*a^76*x^76 + 379976498200746407269569615183607643504640000*a^75 
*x^75 + 28498237365055980545217721138770573262848000000*a^74*x^74 + 210886 
9565014142560346111364269022421450752000000*a^73*x^73 + 153947478246032406 
905266129591638636765904896000000*a^72*x^72 + 1108421843371433329717916133 
0597981847145152512000000*a^71*x^71 + 786979508793717664099720454472456711 
147305828352000000*a^70*x^70 + 5508856561556023648698043181307196978031140 
7984640000000*a^69*x^69 + 380111102747365631760164979510196591484148715094 
0160000000*a^68*x^68 + 258475549868208629596912186066933682209221126263930 
880000000*a^67*x^67 + 1731786184116997818299311646648455670801781545968...
 

Sympy [F(-1)]

Timed out. \[ \int x^{100} \Gamma (-2,a x) \, dx=\text {Timed out} \] Input:

integrate(x**100*uppergamma(-2,a*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{100} \Gamma (-2,a x) \, dx=\int { x^{100} \Gamma \left (-2, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(-2,a*x),x, algorithm="maxima")
 

Output:

integrate(x^100*gamma(-2, a*x), x)
 

Giac [F]

\[ \int x^{100} \Gamma (-2,a x) \, dx=\int { x^{100} \Gamma \left (-2, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(-2,a*x),x, algorithm="giac")
 

Output:

integrate(x^100*gamma(-2, a*x), x)
 

Mupad [B] (verification not implemented)

Time = 2.37 (sec) , antiderivative size = 814, normalized size of antiderivative = 32.56 \[ \int x^{100} \Gamma (-2,a x) \, dx=\text {Too large to display} \] Input:

int((x^98*expint(3, a*x))/a^2,x)
 

Output:

-(x^99*(exp(-a*x)*(1/(a*x) + 98/(a^2*x^2) + 9506/(a^3*x^3) + 912576/(a^4*x 
^4) + 86694720/(a^5*x^5) + 8149303680/(a^6*x^6) + 757885242240/(a^7*x^7) + 
 69725442286080/(a^8*x^8) + 6345015248033280/(a^9*x^9) + 57105137232299520 
0/(a^10*x^10) + 50823572136746572800/(a^11*x^11) + 4472474348033698406400/ 
(a^12*x^12) + 389105268278931761356800/(a^13*x^13) + 334630530719881314766 
84800/(a^14*x^14) + 2844359511118991175518208000/(a^15*x^15) + 23892619893 
3995258743529472000/(a^16*x^16) + 19830874511521606475712946176000/(a^17*x 
^17) + 1626131709944771731008461586432000/(a^18*x^18) + 131716668505526510 
211685388500992000/(a^19*x^19) + 10537333480442120816934831080079360000/(a 
^20*x^20) + 832449344954927544537851655326269440000/(a^21*x^21) + 64931048 
906484348473952429115449016320000/(a^22*x^22) + 49996907657992948324943370 
41889574256640000/(a^23*x^23) + 379976498200746407269569615183607643504640 
000/(a^24*x^24) + 28498237365055980545217721138770573262848000000/(a^25*x^ 
25) + 2108869565014142560346111364269022421450752000000/(a^26*x^26) + 1539 
47478246032406905266129591638636765904896000000/(a^27*x^27) + 110842184337 
14333297179161330597981847145152512000000/(a^28*x^28) + 786979508793717664 
099720454472456711147305828352000000/(a^29*x^29) + 55088565615560236486980 
431813071969780311407984640000000/(a^30*x^30) + 38011110274736563176016497 
95101965914841487150940160000000/(a^31*x^31) + 258475549868208629596912186 
066933682209221126263930880000000/(a^32*x^32) + 17317861841169978182993...
 

Reduce [F]

\[ \int x^{100} \Gamma (-2,a x) \, dx=\frac {\int \mathit {ei} \left (3, a x \right ) x^{98}d x}{a^{2}} \] Input:

int(x^98/a^2*Ei(3,a*x),x)
 

Output:

int(ei(3,a*x)*x**98,x)/a**2